The relative monoidal center and tensor products of monoidal categories

This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. It is shown that there exists a monoidal structure on the relative tensor product of two augmented monoidal categories which is Morita dual to a relative version of the monoidal center. In examples, a category of locally finite weight modules over a quantized enveloping algebra is equivalent to the relative monoidal center of modules over its Borel part. A similar result holds for small quantum groups, without restricting to locally finite weight modules. More generally, for modules over bialgebras inside a braided monoidal category, the relative center is shown to be equivalent to the category of Yetter–Drinfeld modules inside the braided category. If the braided category is given by modules over a quasitriangular Hopf algebra, then the relative center corresponds to modules over a braided version of the Drinfeld double (i.e. the double bosonization in the sense of Majid) which are locally finite for the action of the dual.

Universal K-matrix for quantum symmetric pairs

Let {{\mathfrak{g}}} be a symmetrizable Kac–Moody algebra and let {{U_{q}(\mathfrak{g})}} denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras {{B_{\mathbf{c},\mathbf{s}}}} of {{U_{q}(\mathfrak{g})}} have a universal K-matrix if {{\mathfrak{g}}} is of finite type. By a universal K-matrix for {{B_{\mathbf{c},\mathbf{s}}}} we mean an element in a completion of {{U_{q}(\mathfrak{g})}} which commutes with {{B_{\mathbf{c},\mathbf{s}}}} and provides solutions of the reflection equation in all integrable {{U_{q}(\mathfrak{g})}} -modules in category {{\mathcal{O}}} . The construction of the universal K-matrix for {{B_{\mathbf{c},\mathbf{s}}}} bears significant resemblance to the construction of the universal R-matrix for {{U_{q}(\mathfrak{g})}} . Most steps in the construction of the universal K-matrix are performed in the general Kac–Moody setting. In the late nineties T. tom Dieck and R. Häring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.

Gröbner-Shirshov Basis and Minimal Projective Resolution of Uq+(B2)

In this paper, by using the Anick resolution and Gröbner-Shirshov basis for quantized enveloping algebra of type B2, we compute the minimal projective resolution of the trivial module of [Formula: see text], and as an application we compute the global dimension of [Formula: see text].

Factorizable Module Algebras

The aim of this paper is to introduce and study a large class of $\mathfrak{g}$-module algebras that we call factorizable by generalizing the Gauss factorization of square or rectangular matrices. This class includes coordinate algebras of corresponding reductive groups G, their parabolic subgroups, basic affine spaces, and many others. It turns out that products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any $\mathfrak{g}$-module algebra. We also have quantum versions of all these constructions in the category of $U_{q}(\mathfrak{g})$-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra $U_{q}(\mathfrak{g}^{\ast })$ of the dual Lie bialgebra $\mathfrak{g}^{\ast }$ of $\mathfrak{g}$.

Gröbner–Shirshov pair of irreducible modules over quantized enveloping algebra of type An

In this paper, first, we give a Gröbner–Shirshov pair of finite-dimensional irreducible module [Formula: see text] over [Formula: see text] the quantized enveloping algebra of type [Formula: see text] by using the double free module method and the known Gröbner–Shirshov basis of [Formula: see text] Then, by specializing a suitable version of [Formula: see text] at [Formula: see text] we get a Gröbner–Shirshov basis of [Formula: see text] and get a Gröbner–Shirshov pair for the finite-dimensional irreducible module [Formula: see text] over [Formula: see text].

Homomorphisms between standard modules over finite-type KLR algebras

Khovanov–Lauda–Rouquier (KLR) algebras of finite Lie type come with families of standard modules, which under the Khovanov–Lauda–Rouquier categorification correspond to PBW bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of a standard module are injective. We present applications to the extensions between standard modules and modular representation theory of KLR algebras.

The universal sl2 invariant and Milnor invariants

The universal [Formula: see text] invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the [Formula: see text]-adic completed tensor powers of the quantized enveloping algebra of [Formula: see text]. In this paper, we exhibit explicit relationships between the universal [Formula: see text] invariant and Milnor invariants, which are classical invariants generalizing the linking number, providing some new topological insight into quantum invariants. More precisely, we define a reduction of the universal [Formula: see text] invariant, and show how it is captured by Milnor concordance invariants. We also show how a stronger reduction corresponds to Milnor link-homotopy invariants. As a byproduct, we give explicit criterions for invariance under concordance and link-homotopy of the universal [Formula: see text] invariant, and in particular for sliceness. Our results also provide partial constructions for the still-unknown weight system of the universal [Formula: see text] invariant.

Quantum groups via cyclic quiver varieties I

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.